CN114997020B - Multi-degree-of-freedom vehicle dynamic response-based contact point inversion algorithm - Google Patents
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Abstract
The invention discloses a contact point inversion algorithm based on multi-degree-of-freedom vehicle dynamic response, and belongs to the field of railway engineering health detection. The invention is based on a simplified physical model of a multi-degree-of-freedom vehicle track bridge coupling system, the adopted vehicle model is a six-degree-of-freedom system, a multi-degree-of-freedom vehicle lower track-bridge interaction theory is established, and analysis correlation of track, bridge structure and vehicle body response is ascertained; the closed analytical solution of the multi-contact point response of the vehicle and the track in the six-degree-of-freedom vehicle-track-bridge coupling system is deduced, and the correctness of the algorithm is verified by MATLAB modeling.
Description
Technical Field
The invention relates to the field of railway engineering health monitoring, in particular to a contact point inversion algorithm based on multi-degree-of-freedom vehicle dynamic response.
Background
Along with the continuous improvement of bridge span and train running speed, researchers have made a great deal of work on the dynamic characteristics between the bridge and the train and applied to practical engineering in order to ensure the safety and the comfort of the bridge running performance and the train running, however, a track system which is an important component of a railway is often ignored, and from the prior literature, the track system has a large coupling response to the axle, so that the analysis of the train-track-bridge as an integral system has become a research trend of the characteristics of the railway system in recent years. Most research has been carried over with single degree of freedom vehicle models, and few have studied the coupling and rotation between the front and rear axles of the vehicle, which is quite different from the actual engineering application of trains.
Therefore, there is a need to develop a method of acquiring the contact point response between different wheel sets and rails.
Disclosure of Invention
The invention aims to provide a contact point inversion algorithm based on the dynamic response of a vehicle with multiple degrees of freedom, so as to solve the problems in the prior art.
The technical scheme adopted for realizing the purpose of the invention is that the contact point inversion algorithm based on the dynamic response of the vehicle with multiple degrees of freedom comprises the following steps:
1) Based on a simplified physical model of the multi-degree-of-freedom vehicle-track-bridge coupling system, establishing a motion equation of the multi-degree-of-freedom vehicle system;
2) Analyzing a bogie motion equation in a vehicle system, and calculating a closed analysis solution general expression of the four wheel and track contact points;
3) Obtaining a general expression of the contact point response in a discrete form according to the contact point response expression in the continuous time state in the step 2);
4) Based on a simplified physical model of the multi-degree-of-freedom vehicle track bridge coupling system and a vehicle-track-bridge coupling unit model, establishing a motion equation of the vehicle-track-bridge coupling system;
5) Establishing a finite element model of the vehicle-track-bridge coupling system according to the motion equation of the vehicle-track-bridge coupling unit obtained in the step 4);
6) And calculating the response of the whole vehicle-track-bridge coupling system by using a Newmark-beta method.
Further, in the simplified physical model of the multi-degree-of-freedom vehicle-track system described in step 1), the vehicle is simplified into a six-degree-of-freedom model consisting of one vehicle body, two bogies and four wheel pairs, each component being a rigid member;
Wherein, the car body is simplified into a rigid beam with the length of 2l c, the mass of m c and the mass moment of inertia of J c; the bogie is simplified into a rigid beam with the length of 2l t, the mass of m t and the mass moment of inertia of J t, and t 1 and t 2 respectively represent a front bogie and a rear bogie; the two bogies are connected with the vehicle body through a secondary suspension system, the secondary suspension system consists of a spring I and a damper I, the rigidity of the spring I is K s, and the damping of the damper I is C s; the elastic connection system between the four wheel sets and the bogie is a suspension system, the suspension system consists of a spring II and a damper II, the rigidity of the spring II is K p, the damping of the damper II is C p, the four wheel sets are rigidly simplified, the mass of the four wheel sets is m w1、mw2、mw3 and m w4 respectively, the wheel sets with mass of m w1 and m w2 are connected with the front bogie, and the wheel sets with mass of m w3 and m w4 are connected with the rear bogie.
Further, step 2) comprises the following sub-steps:
2-1) carrying out stress analysis on front and rear bogies of a vehicle, and respectively establishing a vertical motion equation and a rotation balance equation of the vehicle body, the front bogie and the rear bogie by using the principle of D' Alembert:
Vertical motion equation of the vehicle body:
Wherein: y c represents the vertical displacement of the vehicle body, Representing the derivation of the vertical displacement of the vehicle body to time,/>Represents the vertical displacement of the vehicle body and the time is subjected to secondary conduction, theta c represents the in-plane angle of the vehicle body,/>Represents the derivation of the in-plane angle of the vehicle body over time, y t1 represents the vertical displacement of the front bogie, and is/isRepresenting the front bogie vertical displacement deriving time, y t2 representing the rear bogie vertical displacement,/>The vertical displacement of the rear bogie is expressed and the time is derived;
Vehicle body rotation balance equation:
Wherein: the second conduction of the vehicle in-plane angle to time is shown;
Front truck vertical equation of motion:
Wherein: represents the vertical displacement of the front bogie and the time is subjected to secondary guidance, and theta t1 represents the in-plane angle of the front bogie, and is/are Represents the in-plane angle of the front bogie for time derivation, u c1 represents the vertical displacement of the contact point of the first wheel set and the track,/>Representing the derivative of the vertical displacement of the first wheel set with respect to the rail contact point, u c2 represents the vertical displacement of the second wheel set with respect to the rail contact point,Representing the vertical displacement of the contact point of the second wheel set and the track to derive time;
Front truck rotation balance equation:
Wherein: the in-plane angle of the front bogie is represented to calculate the secondary guide of time;
Vertical motion equation of rear bogie:
Wherein: Represents the vertical displacement of the rear bogie to be conducted secondarily with respect to time, and theta t2 represents the in-plane angle of the rear bogie, and is/are Represents the derivation of the in-plane angle of the rear bogie over time, u c3 represents the vertical displacement of the third wheel pair at the contact point with the track,/>Representing the derivative of the vertical displacement of the third wheel pair with respect to the time, u c4 represents the vertical displacement of the fourth wheel pair with respect to the rail,The vertical displacement of the contact point of the fourth wheel pair and the track is expressed, and the time is derived;
rear truck rotation balance equation:
Wherein: the in-plane angle of the rear bogie is represented to calculate the secondary conduction of time;
2-2) the formulas (3), (4), (5) and (6) are obtained by the term-shifting arrangement, respectively:
By l t × (7) + (8), the first wheel set to track contact point response equation is obtained:
taking the second derivative of equation (11) yields a first order linear differential equation for the first wheel pair to track contact acceleration response:
G c1 in the formula is obtained by field test, and the expression is:
by l t × (7) - (8), the second wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (14) yields a first order linear differential equation for the second wheel pair to track contact acceleration response:
wherein G c2 is:
by l t × (9) + (10), the third wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (17) yields a first order linear differential equation for the third wheel pair versus orbital contact point acceleration response:
wherein G c3 is:
By l t × (9) - (10), the fourth pair-to-track contact point response equation is obtained:
taking the second derivative of equation (20) yields a first order linear differential equation for the fourth pair-to-track contact acceleration response:
Wherein G c4 is:
wherein the vertical acceleration response of the vehicle in equations (13), (16), (19) and (22) And rotational acceleration response/>The first derivative and the second derivative of the acceleration response are calculated by a central difference method, wherein the first derivative and the second derivative are obtained by measuring sensors arranged at various parts of the vehicle and are as follows:
wherein j is the number of sampling points of the vehicle response, and Δt represents the sampling time interval;
Solving differential equations (12), (15), (18) and (21) for each contact point, assuming that each contact point response vibrates from t=0, four contact point responses in the model are obtained The general expression of (2) is:
Further, in step 3), the contact point acceleration response of equation (27) is expressed in discrete form as:
Wherein G ci|j is the value of the j-th sampling point of G ci in formulas (13), (16), (19) and (22).
Further, in the motion equation of the vehicle-track-bridge coupling system established in the step 4), four wheel pairs are rigid models.
Further, step 6) is followed by the steps of: and (3) comparing the wheel-rail contact point response obtained in the step (3) with the wheel-rail contact point response obtained in the step (6) by a finite element method, and verifying the correctness and accuracy of the obtained wheel-rail contact point response closed-form analytic solution general expression.
The invention has the beneficial effects that:
1. The method establishes a multi-degree-of-freedom vehicle lower track-bridge interaction theory, and confirms the analysis correlation of the track, the bridge structure and the vehicle body response;
2. The method has a strict theoretical basis, is based on a six-degree-of-freedom vehicle track bridge coupling system, derives an acceleration response closed analysis solution general expression of wheel/track multi-contact points by utilizing vehicle body response, and provides a mathematical basis for subsequent research;
3. the simplified physical model of the vehicle body adopted by the method is a six-degree-of-freedom system which is more close to the condition of actual operation trains, more detail influence is considered compared with the traditional single-degree-of-freedom model and double-degree-of-freedom model, and the wheel/gauge multi-contact point response is reversely calculated according to the acquired vehicle body response signals, so that the accuracy of the identified orbit modulus is higher;
4. the vehicle and track multi-contact point response obtained by the method is obtained only by deduction according to the mechanical balance condition of each part of the structure of the vehicle, so that the vehicle and track multi-contact point response is not limited by the working performance of each component in the railway system, and the theoretical model is suitable for each type of railway system;
5. In the method, the data processing of the actually measured signals of each part of the train is programmed on the Matlab, so that the integration of signal real-time transmission, data analysis and result output is realized, and the method has the high efficiency of an indirect measurement technology.
Drawings
FIG. 1 is a simplified physical model of a multiple degree of freedom vehicle-rail-bridge coupling system;
FIG. 2 is a schematic diagram of a vehicle front truck vertical stress analysis;
FIG. 3 is a schematic diagram of a rotational force analysis of a front truck of a vehicle;
FIG. 4 is a diagram of a track-bridge unit model of a vehicle with a wheel set acting on the track;
fig. 5 is a graph of a theoretical solution to a finite element solution for a multi-round-rail contact point response.
Detailed Description
The present invention is further described below with reference to examples, but it should not be construed that the scope of the above subject matter of the present invention is limited to the following examples. Various substitutions and alterations are made according to the ordinary skill and familiar means of the art without departing from the technical spirit of the invention, and all such substitutions and alterations are intended to be included in the scope of the invention.
Example 1:
The embodiment discloses a contact point inversion algorithm based on multi-degree-of-freedom vehicle dynamic response, which comprises the following steps:
1) Based on a simplified physical model of the multi-degree-of-freedom vehicle-track-bridge coupling system, establishing a motion equation of the multi-degree-of-freedom vehicle system; specifically, in the simplified physical model of the multi-degree-of-freedom vehicle-track system, the vehicle is simplified into a six-degree-of-freedom model consisting of a vehicle body, two bogies and four wheel pairs, and each component is a rigid member;
Referring to fig. 1, the vehicle body is simplified into a rigid beam with a length of 2l c, a mass of m c and a mass moment of inertia of J c; the bogie is simplified into a rigid beam with the length of 2l t, the mass of m t and the mass moment of inertia of J t, and t 1 and t 2 respectively represent a front bogie and a rear bogie; the two bogies are connected with the vehicle body through a secondary suspension system, the secondary suspension system consists of a spring I and a damper I, the rigidity of the spring I is K s, and the damping of the damper I is C s; the elastic connection system between the four wheel sets and the bogie is a suspension system, the suspension system consists of a spring II and a damper II, the rigidity of the spring II is K p, the damping of the damper II is C p, the four wheel sets are rigidly simplified, the mass of the four wheel sets is m w1、mw2、mw3 and m w4 respectively, the wheel sets with mass of m w1 and m w2 are connected with the front bogie, and the wheel sets with mass of m w3 and m w4 are connected with the rear bogie;
2) Analyzing a bogie motion equation in a vehicle system, and calculating a closed analysis solution general expression of the four wheel and track contact points; the method specifically comprises the following sub-steps:
2-1) carrying out stress analysis on front and rear bogies of a vehicle, referring to fig. 2 and 3, respectively, a vertical stress analysis schematic diagram of the front bogie of the vehicle and a rotation stress analysis schematic diagram of the front bogie of the vehicle, and respectively establishing a vertical motion equation and a rotation balance equation of the vehicle body, the front bogie and the rear bogie by using the principle of D' Alembert:
Vertical motion equation of the vehicle body:
Wherein: y c represents the vertical displacement of the vehicle body, Representing the derivation of the vertical displacement of the vehicle body to time,/>Represents the vertical displacement of the vehicle body and the time is subjected to secondary conduction, theta c represents the in-plane angle of the vehicle body,/>Represents the derivation of the in-plane angle of the vehicle body over time, y t1 represents the vertical displacement of the front bogie, and is/isRepresenting the front bogie vertical displacement deriving time, y t2 representing the rear bogie vertical displacement,/>The vertical displacement of the rear bogie is expressed and the time is derived;
Vehicle body rotation balance equation:
Wherein: the second conduction of the vehicle in-plane angle to time is shown;
Front truck vertical equation of motion:
Wherein: represents the vertical displacement of the front bogie and the time is subjected to secondary guidance, and theta t1 represents the in-plane angle of the front bogie, and is/are Represents the in-plane angle of the front bogie for time derivation, u c1 represents the vertical displacement of the contact point of the first wheel set and the track,/>Representing the derivative of the vertical displacement of the first wheel set with respect to the rail contact point, u c2 represents the vertical displacement of the second wheel set with respect to the rail contact point,Representing the vertical displacement of the contact point of the second wheel set and the track to derive time;
Front truck rotation balance equation:
Wherein: the in-plane angle of the front bogie is represented to calculate the secondary guide of time;
Vertical motion equation of rear bogie:
Wherein: Represents the vertical displacement of the rear bogie to be conducted secondarily with respect to time, and theta t2 represents the in-plane angle of the rear bogie, and is/are Represents the derivation of the in-plane angle of the rear bogie over time, u c3 represents the vertical displacement of the third wheel pair at the contact point with the track,/>Representing the derivative of the vertical displacement of the third wheel pair with respect to the time, u c4 represents the vertical displacement of the fourth wheel pair with respect to the rail,The vertical displacement of the contact point of the fourth wheel pair and the track is expressed, and the time is derived;
rear truck rotation balance equation:
Wherein: the in-plane angle of the rear bogie is represented to calculate the secondary conduction of time;
2-2) the formulas (3), (4), (5) and (6) are obtained by the term-shifting arrangement, respectively:
By l t × (7) + (8), the first wheel set to track contact point response equation is obtained:
taking the second derivative of equation (11) yields a first order linear differential equation for the first wheel pair to track contact acceleration response:
G c1 in the formula is obtained by field test, and the expression is:
by l t × (7) - (8), the second wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (14) yields a first order linear differential equation for the second wheel pair to track contact acceleration response:
wherein G c2 is:
by l t × (9) + (10), the third wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (17) yields a first order linear differential equation for the third wheel pair versus orbital contact point acceleration response:
wherein G c3 is:
By l t × (9) - (10), the fourth pair-to-track contact point response equation is obtained:
taking the second derivative of equation (20) yields a first order linear differential equation for the fourth pair-to-track contact acceleration response:
Wherein G c4 is:
wherein the vertical acceleration response of the vehicle in equations (13), (16), (19) and (22) And rotational acceleration response/>The first derivative and the second derivative of the acceleration response are calculated by a central difference method, wherein the first derivative and the second derivative are obtained by measuring sensors arranged at various parts of the vehicle and are as follows:
wherein j is the number of sampling points of the vehicle response, and Δt represents the sampling time interval;
Solving differential equations (12), (15), (18) and (21) for each contact point, assuming that each contact point response vibrates from t=0, four contact point responses in the model are obtained The general expression of (2) is:
3) The multi-contact point response expression obtained in the step 2) is suitable for the condition of continuous time, in actual engineering, a fixed sampling time interval is required to be set, vibration data of each component part of the vehicle is obtained through six sensor records installed at each part of the vehicle, the data processing of actual measurement signals of each part of the train is programmed on Matlab, the data are discrete in nature, and the multi-contact point response expression under the condition of continuous time is not suitable for the multi-contact point response expression; step 3) is performed to obtain a general expression of a discrete form of multi-contact-point response, and meanwhile, the expression of the discrete form only relates to the derivative of the acceleration of each component part of the vehicle, and no integral related term exists, so that the problem of baseline drift does not exist. Specific:
since vibration data obtained by the six sensor recordings mounted at each part of the vehicle are discrete in nature, the contact point acceleration response of equation (27) is expressed in discrete form as:
Wherein G ci|j is the value of the j-th sampling point of G ci in formulas (13), (16), (19) and (22).
From equations (13), (16), (19) and (22), it is found that G ci (t) is related only to the vibration response, performance of the vehicle, and is not affected by the vibration response and performance of the track and bridge. The multi-contact point response of the vehicle to the track in the formula (28) is obtained only by deducting the mechanical balance condition of each part of the structure of the vehicle, and is not limited by the working performance of each component in the railway system, and the theoretical model is applicable to each type of railway system.
4) Based on a simplified physical model of the multi-degree-of-freedom vehicle track bridge coupling system and a vehicle-track-bridge coupling unit model, a motion equation of the vehicle-track-bridge coupling system is established, four wheel pairs are rigid models, deformation is not considered, and the wheel pair quality is considered into track quality when finite element modeling is carried out, so that along with the movement of the vehicle on a double beam, the quality matrix of a track unit corresponding to the position where the vehicle runs changes; FIG. 4 is a diagram of a track-bridge unit model of a vehicle with a wheel set acting on the track;
In the simplified physical model of the multi-degree-of-freedom vehicle track bridge coupling system in fig. 1, the measuring vehicle moves on a simply supported double beam with a span of L at a speed v, the running vehicle is used as a load to excite a track, namely a bridge, to generate vibration, and vice versa, the vibrating track/bridge can cause vertical vibration of a moving vehicle. The mass of the rail and the bridge is m r、mb respectively, the span is L, the constant section Bernoulli-Euler beam is used for simulating the influence of rail fasteners, sleepers and railway ballasts between two simply supported beams by using a spring buffer unit equivalent system with a uniform stiffness coefficient theta and a damping coefficient C, and the influence is called an intermediate elastic layer; e r is the elastic modulus of the track, I r is the section moment of inertia of the track, E b is the elastic modulus of the bridge, and I b is the section moment of inertia of the bridge.
For a more practical engineering, the model constructed here is a generalized vehicle-rail-bridge interaction unit taking into account vehicle/rail/bridge damping and road surface roughness, as shown in fig. 3, the rail and bridge units are each simplified to be planar beam units of one-dimensional two nodes, each unit node taking into account only vertical displacement (v i) and in-plane angleIn the local unit coordinate system x eoeye, track and bridge units with the length of l e are established, and the bridge units and the track units are connected with each other by uniformly distributed springs-damping units. In the method of the invention, the relevant modulus analysis is mainly carried out through the contact point response of the vehicle and the track, so that the coupling condition of the vehicle and the track/bridge is only shown by the contact of a single wheel pair and the track surface in fig. 3, the relative position of the wheel pair and the track is denoted by xi ih, and r i is a relevant function for describing the roughness of the track surface. Based on the energy principle, the motion equation of the vehicle-rail-bridge coupling unit considering the surface roughness of the rail can be established as follows:
Wherein [ M ], [ C ] and [ K ] are mass, damping and stiffness matrices of the unit, respectively; subscripts v, r, and b represent relevant parameters of the vehicle, track, and bridge, respectively; vr and rv represent coupling matrices between the vehicle and the track; br and rb represent coupling matrices between the track and the bridge, respectively; { q } represents cell node displacement; { F (t) } is the unit load.
For the vehicle model simplified into six degrees of freedom in the invention, the displacement matrix of the vehicle with multiple degrees of freedom is as follows:
qv＝[yc θc yt1 θt1 yt2 θt2] (30)
5) Establishing a finite element model of the vehicle-track-bridge coupling system according to the motion equation of the vehicle-track-bridge coupling unit obtained in the step 4); by assembling all track units (including the track units directly acted by the vehicle body and other common track units) and bridge units, the whole motion equation of the whole VRB system can be established.
6) Calculating the response of the whole vehicle-track-bridge coupling system by using a Newmark-beta method (beta=0.25, gamma=0.5);
7) And (3) comparing the wheel-rail contact point response obtained in the step (3) with the wheel-rail contact point response obtained in the step (6) by a finite element method, and verifying the correctness and accuracy of the obtained wheel-rail contact point response closed-form analytic solution general expression.
Specifically, in order to verify the correctness of the method of the present invention, the embodiment uses MATLAB to perform finite element modeling and calculation based on the theoretical derivation, verifies the correctness of the closed analytical solution expression of the wheel/rail multi-contact point acceleration response, performs finite element modeling on MATLAB to obtain a track-bridge simply supported double beam with the span of 32m, and simulates the condition that a six-degree-of-freedom train model with acceleration, displacement and speed sensors passes through the track bridge. In MATLAB calculation, the train passes through the track bridge at a running speed of 10m/s, and physical parameters of the used vehicle, track and bridge are respectively given in tables 1 and 2, and in the finite element analysis simulation process, the track is divided into 32 units, and the length of each unit is 1m. When verifying a closed solution of the multi-contact point response of the vehicle and the track, the vibration response is represented by adopting the first 5-order eigenmode, and the mode can be proved to represent more accurate vibration response in the previous research of the team. When solving the overall motion equation, the time step takes 0.0001s.
Table 1 vehicle physical parameter values
Table 2 physical parameter values for track-bridge
Finally, the finite element solution of the acceleration response of each contact point of the vehicle and the track is compared with the back calculation result of the formula (28) and is shown in fig. 5, wherein fig. 5 (a), 5 (b), 5 (c) and 5 (d) are the acceleration responses of the contact points of the first wheel pair, the second wheel pair, the third wheel pair and the fourth wheel pair of the vehicle and the track respectively, the finite element solution of each contact point is well matched with the back calculation result based on the vehicle response, the correctness of theoretical derivation in the embodiment is illustrated, and the closed-loop analysis solution of the multi-contact point response of the vehicle/track can be fully verified.
Example 2:
The embodiment discloses a contact point inversion algorithm based on multi-degree-of-freedom vehicle dynamic response, which comprises the following steps:
1) Based on a simplified physical model of the multi-degree-of-freedom vehicle-track-bridge coupling system, establishing a motion equation of the multi-degree-of-freedom vehicle system;
2) Analyzing a bogie motion equation in a vehicle system, and calculating a closed analysis solution general expression of the four wheel and track contact points;
3) Obtaining a general expression of the contact point response in a discrete form according to the contact point response expression in the continuous time state in the step 2);
4) Based on a simplified physical model of the multi-degree-of-freedom vehicle track bridge coupling system and a vehicle-track-bridge coupling unit model, establishing a motion equation of the vehicle-track-bridge coupling system;
5) Establishing a finite element model of the vehicle-track-bridge coupling system according to the motion equation of the vehicle-track-bridge coupling unit obtained in the step 4);
6) And calculating the response of the whole vehicle-track-bridge coupling system by using a Newmark-beta method.
Example 3:
The main steps of the embodiment are the same as those of the embodiment 2, and further, in the simplified physical model of the vehicle-track system with multiple degrees of freedom in the step 1), the vehicle is simplified into a six-degree-of-freedom model consisting of a vehicle body, two bogies and four wheel pairs, and each component is a rigid member;
Wherein, the car body is simplified into a rigid beam with the length of 2l c, the mass of m c and the mass moment of inertia of J c; the bogie is simplified into a rigid beam with the length of 2l t, the mass of m t and the mass moment of inertia of J t, and t 1 and t 2 respectively represent a front bogie and a rear bogie; the two bogies are connected with the vehicle body through a secondary suspension system, the secondary suspension system consists of a spring I and a damper I, the rigidity of the spring I is K s, and the damping of the damper I is C s; the elastic connection system between the four wheel sets and the bogie is a suspension system, the suspension system consists of a spring II and a damper II, the rigidity of the spring II is K p, the damping of the damper II is C p, the four wheel sets are rigidly simplified, the mass of the four wheel sets is m w1、mw2、mw3 and m w4 respectively, the wheel sets with mass of m w1 and m w2 are connected with the front bogie, and the wheel sets with mass of m w3 and m w4 are connected with the rear bogie.
Example 4:
the main steps of this embodiment are the same as those of embodiment 3, and further, step 2) includes the following sub-steps:
2-1) carrying out stress analysis on front and rear bogies of a vehicle, and respectively establishing a vertical motion equation and a rotation balance equation of the vehicle body, the front bogie and the rear bogie by using the principle of D' Alembert:
Vertical motion equation of the vehicle body:
Wherein: y c represents the vertical displacement of the vehicle body, Representing the derivation of the vertical displacement of the vehicle body to time,/>Represents the vertical displacement of the vehicle body and the time is subjected to secondary conduction, theta c represents the in-plane angle of the vehicle body,/>Represents the derivation of the in-plane angle of the vehicle body over time, y t1 represents the vertical displacement of the front bogie, and is/isRepresenting the front bogie vertical displacement deriving time, y t2 representing the rear bogie vertical displacement,/>The vertical displacement of the rear bogie is expressed and the time is derived;
Vehicle body rotation balance equation:
Wherein: the second conduction of the vehicle in-plane angle to time is shown;
Front truck vertical equation of motion:
Wherein: represents the vertical displacement of the front bogie and the time is subjected to secondary guidance, and theta t1 represents the in-plane angle of the front bogie, and is/are Represents the in-plane angle of the front bogie for time derivation, u c1 represents the vertical displacement of the contact point of the first wheel set and the track,/>Representing the derivative of the vertical displacement of the first wheel set with respect to the rail contact point, u c2 represents the vertical displacement of the second wheel set with respect to the rail contact point,Representing the vertical displacement of the contact point of the second wheel set and the track to derive time;
Front truck rotation balance equation:
Wherein: the in-plane angle of the front bogie is represented to calculate the secondary guide of time;
Vertical motion equation of rear bogie:
Wherein: Represents the vertical displacement of the rear bogie to be conducted secondarily with respect to time, and theta t2 represents the in-plane angle of the rear bogie, and is/are Represents the derivation of the in-plane angle of the rear bogie over time, u c3 represents the vertical displacement of the third wheel pair at the contact point with the track,/>Representing the derivative of the vertical displacement of the third wheel pair with respect to the time, u c4 represents the vertical displacement of the fourth wheel pair with respect to the rail,The vertical displacement of the contact point of the fourth wheel pair and the track is expressed, and the time is derived;
rear truck rotation balance equation:
Wherein: the in-plane angle of the rear bogie is represented to calculate the secondary conduction of time;
2-2) the formulas (3), (4), (5) and (6) are obtained by the term-shifting arrangement, respectively:
By l t × (7) + (8), the first wheel set to track contact point response equation is obtained:
taking the second derivative of equation (11) yields a first order linear differential equation for the first wheel pair to track contact acceleration response:
G c1 in the formula is obtained by field test, and the expression is:
by l t × (7) - (8), the second wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (14) yields a first order linear differential equation for the second wheel pair to track contact acceleration response:
wherein G c2 is:
by l t × (9) + (10), the third wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (17) yields a first order linear differential equation for the third wheel pair versus orbital contact point acceleration response:
wherein G c3 is:
By l t × (9) - (10), the fourth pair-to-track contact point response equation is obtained:
taking the second derivative of equation (20) yields a first order linear differential equation for the fourth pair-to-track contact acceleration response:
/>
Wherein G c4 is:
wherein the vertical acceleration response of the vehicle in equations (13), (16), (19) and (22) And rotational acceleration response/>The first derivative and the second derivative of the acceleration response are calculated by a central difference method, wherein the first derivative and the second derivative are obtained by measuring sensors arranged at various parts of the vehicle and are as follows:
wherein j is the number of sampling points of the vehicle response, and Δt represents the sampling time interval;
Solving differential equations (12), (15), (18) and (21) for each contact point, assuming that each contact point response vibrates from t=0, four contact point responses in the model are obtained The general expression of (2) is:
Example 5:
the main steps of this embodiment are the same as those of embodiment 4, and further, in step 3), the contact point acceleration response of the formula (27) is expressed in a discrete form as:
Wherein G ci|j is the value of the j-th sampling point of G ci in formulas (13), (16), (19) and (22).
Example 6:
the main steps of the embodiment are the same as those of the embodiment 2, and further, in the motion equation of the vehicle-track-bridge coupling system established in the step 4), four wheel pairs are rigid models.
Example 7:
The main steps of this embodiment are the same as those of embodiment 2, and further, the following steps are further provided after step 6): and (3) comparing the wheel-track contact point response obtained in the step (3) with the wheel-track contact point response obtained in the step (6) by a finite element method, and verifying the correctness and accuracy of the obtained wheel-track contact point response closed-form analytical solution general expression.
Claims (4)
1. The contact point inversion algorithm based on the vehicle dynamic response with multiple degrees of freedom is characterized by comprising the following steps:
1) Based on a simplified physical model of the multi-degree-of-freedom vehicle-track-bridge coupling system, establishing a motion equation of the multi-degree-of-freedom vehicle system; in the simplified physical model of the vehicle-track system with multiple degrees of freedom, the vehicle is simplified into a six-degree-of-freedom model consisting of a vehicle body, two bogies and four wheel pairs, and each component is a rigid member;
Wherein, the car body is simplified into a rigid beam with the length of 2l c, the mass of m c and the mass moment of inertia of J c; the bogie is simplified into a rigid beam with the length of 2l t, the mass of m t and the mass moment of inertia of J t, and t 1 and t 2 respectively represent a front bogie and a rear bogie; the two bogies are connected with the vehicle body through a secondary suspension system, the secondary suspension system consists of a spring I and a damper I, the rigidity of the spring I is K s, and the damping of the damper I is C s; the elastic connection system between the four wheel sets and the bogie is a suspension system, the suspension system consists of a spring II and a damper II, the rigidity of the spring II is K p, the damping of the damper II is C p, the four wheel sets are rigidly simplified, the mass of the four wheel sets is m w1、mw2、mw3 and m w4 respectively, the wheel sets with mass of m w1 and m w2 are connected with the front bogie, and the wheel sets with mass of m w3 and m w4 are connected with the rear bogie;
2) Analyzing a bogie motion equation in a vehicle system, and calculating a closed analysis solution general expression of the four wheel and track contact points; step 2) comprises the following sub-steps:
2-1) carrying out stress analysis on front and rear bogies of a vehicle, and respectively establishing a vertical motion equation and a rotation balance equation of the vehicle body, the front bogie and the rear bogie by using the principle of D' Alembert:
Vertical motion equation of the vehicle body:
Wherein: y c represents the vertical displacement of the vehicle body, Representing the derivation of the vertical displacement of the vehicle body to time,/>Represents the vertical displacement of the vehicle body and the time is subjected to secondary conduction, theta c represents the in-plane angle of the vehicle body,/>Represents the derivation of the in-plane angle of the vehicle body over time, y t1 represents the vertical displacement of the front bogie, and is/isRepresenting the front bogie vertical displacement deriving time, y t2 representing the rear bogie vertical displacement,/>The vertical displacement of the rear bogie is expressed and the time is derived;
Vehicle body rotation balance equation:
Wherein: the second conduction of the vehicle in-plane angle to time is shown;
Front truck vertical equation of motion:
Wherein: represents the vertical displacement of the front bogie and the time is subjected to secondary guidance, and theta t1 represents the in-plane angle of the front bogie, and is/are Represents the in-plane angle of the front bogie for time derivation, u c1 represents the vertical displacement of the contact point of the first wheel set and the track,/>Represents the derivation of the vertical displacement of the contact point of the first wheel set and the track over time, u c2 represents the vertical displacement of the contact point of the second wheel set and the track,/>Representing the vertical displacement of the contact point of the second wheel set and the track to derive time;
Front truck rotation balance equation:
Wherein: the in-plane angle of the front bogie is represented to calculate the secondary guide of time;
Vertical motion equation of rear bogie:
Wherein: Represents the vertical displacement of the rear bogie to be conducted secondarily with respect to time, and theta t2 represents the in-plane angle of the rear bogie, and is/are Represents the derivation of the in-plane angle of the rear bogie over time, u c3 represents the vertical displacement of the third wheel pair at the contact point with the track,/>Represents the derivative of the vertical displacement of the third wheel pair with respect to the time of the contact point of the rail, u c4 represents the vertical displacement of the fourth wheel pair with respect to the rail,/>The vertical displacement of the contact point of the fourth wheel pair and the track is expressed, and the time is derived;
rear truck rotation balance equation:
Wherein: the in-plane angle of the rear bogie is represented to calculate the secondary conduction of time;
2-2) the formulas (3), (4), (5) and (6) are obtained by the term-shifting arrangement, respectively:
By l t × (7) + (8), the first wheel set to track contact point response equation is obtained:
taking the second derivative of equation (11) yields a first order linear differential equation for the first wheel pair to track contact acceleration response:
G c1 in the formula is obtained by field test, and the expression is:
by l t × (7) - (8), the second wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (14) yields a first order linear differential equation for the second wheel pair to track contact acceleration response:
wherein G c2 is:
by l t × (9) + (10), the third wheel pair to track contact point response equation is obtained:
Taking the second derivative of equation (17) yields a first order linear differential equation for the third wheel pair versus orbital contact point acceleration response:
wherein G c3 is:
By l t × (9) - (10), the fourth pair-to-track contact point response equation is obtained:
taking the second derivative of equation (20) yields a first order linear differential equation for the fourth pair-to-track contact acceleration response:
Wherein G c4 is:
wherein the vertical acceleration response of the vehicle in equations (13), (16), (19) and (22) And rotational acceleration responseThe first derivative and the second derivative of the acceleration response are calculated by a central difference method, wherein the first derivative and the second derivative are obtained by measuring sensors arranged at various parts of the vehicle and are as follows:
wherein j is the number of sampling points of the vehicle response, and Δt represents the sampling time interval;
Solving differential equations (12), (15), (18) and (21) for each contact point, assuming that each contact point response vibrates from t=0, four contact point responses in the model are obtained The general expression of (2) is:
3) Obtaining a general expression of the contact point response in a discrete form according to the contact point response expression in the continuous time state in the step 2);
4) Based on a simplified physical model of the multi-degree-of-freedom vehicle track bridge coupling system and a vehicle-track-bridge coupling unit model, establishing a motion equation of the vehicle-track-bridge coupling system;
5) Establishing a finite element model of the vehicle-track-bridge coupling system according to the motion equation of the vehicle-track-bridge coupling unit obtained in the step 4);
6) And calculating the response of the whole vehicle-track-bridge coupling system by using a Newmark-beta method.
2. The multi-degree of freedom vehicle dynamic response based contact point inversion algorithm of claim 1, wherein: in step 3), the contact point acceleration response of equation (27) is expressed in discrete form as:
Wherein G ci|j is the value of the j-th sampling point of G ci in formulas (13), (16), (19) and (22).
3. The multi-degree of freedom vehicle dynamic response based contact point inversion algorithm of claim 1, wherein: in the motion equation of the vehicle-track-bridge coupling system established in the step 4), four wheel pairs are rigid models.
4. The multiple degree of freedom vehicle dynamic response based touch point inversion algorithm of claim 1, wherein step 6) is followed by the steps of: and (3) comparing the wheel-track contact point response obtained in the step (3) with the wheel-track contact point response obtained in the step (6) by a finite element method, and verifying the correctness and accuracy of the obtained wheel-track contact point response closed-form analytical solution general expression.
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